Classical scattering trajectories are known to form a Lagrangian manifold in euclidean phase space,
which allows the classification of local focal points for sufficiently small dimensions. For the case of a short-range
potential, we show that the natural description of focal points at infinity is a Lagrangian manifold in the cotangent
bundle of the sphere and establish the relationship between focal points at infinity and the projection singularities
of that manifold.
HOHBERGER Horst1*, KLEIN Markus2*
. Focal Points at Infinity for Short-Range Scattering Trajectories[J]. Journal of Shanghai Jiaotong University(Science), 2018
, 23(1)
: 146
-157
.
DOI: 10.1007/s12204-018-1920-2
[1] ROBERT D, TAMURA H. Asymptotic behaviour ofscattering amplitudes in semi-classical and low energylimits [J]. Annales-Institut Fourier, 1989, 39(1): 155-192.
[2] DEREZI′NSKI J, G′ ERARD C. Scattering theory ofclassical and quantum N-particle systems [M]. Berlin:Springer-Verlag, 1997.
[3] PROTAS Y N. Quasiclassical asymptotics of the scatteringamplitude for the scattering of a plane wave byinhomogeneities of the medium [J]. Mathematics of theUSSR-Sbornik, 1983, 45(4): 487-506.
[4] VAINBERG B R. Asymptotic methods in equationsof mathematical physics [M]. New York: Gordon andBreach Science Publishrs, 1989.
[5] ARNOL’D V I. Integrals of rapidly oscillating functionsand singularities of projections of Lagrangianmanifolds [J]. Functional Analysis and Its Applications,1972, 6(3): 61-62.
[6] ARNOL’D V I. Normal forms for functions near degeneratecritical points, the Weyl groups of Ak, Dk, Ekand Lagrangian singularities [J]. Functional Anallysisand Its Applications, 1972, 6(4): 254-272.
[7] ARNOL’D V I. Critical points of smooth functions andtheir normal forms [J]. Russian mathematical Surveys,1975, 30(5): 3-65.
[8] REED M, SIMON B. Methods of modern mathematicalphysics III: Scattering theory [M]. New York: AcademicPress, 1979.
